Pressure-induced reconstitution of Fermi surfaces and spin fluctuations in S-substituted FeSe

FeSe is a unique high-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document}Tc iron-based superconductor in which nematicity, superconductivity, and magnetism are entangled with each other in the P-T phase diagram. We performed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{77}$$\end{document}77Se-nuclear magnetic resonance measurements under pressures of up to 3.9 GPa on 12% S-substituted FeSe, in which the complex overlap between the nematicity and magnetism are resolved. A pressure-induced Lifshitz transition was observed at 1.0 GPa as an anomaly of the density of states and as double superconducting (SC) domes accompanied by different types of antiferromagnetic (AF) fluctuations. The low-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\mathrm{c}}$$\end{document}Tc SC dome below 1 GPa is accompanied by strong AF fluctuations, whereas the high-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\mathrm{c}}$$\end{document}Tc SC dome develops above 1 GPa, where AF fluctuations are fairly weak. These results suggest the importance of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{xy}$$\end{document}dxy orbital and its intra-orbital coupling for the high-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\mathrm{c}}$$\end{document}Tc superconductivity.

www.nature.com/scientificreports/ nesting between points X and M with the same orbital can induce a stripe-type magnetic order. When the hole pocket emerges at point M, the shapes of pockets at points Ŵ and X are qualitatively similar to those at ambient pressure, although their size changes monotonically with increasing pressure 21 . The emergence of the d xy hole pocket also enhances the density of states (DOS) (see Fig. 1b), as will be described in detail later. The pressureinduced reconstitution of Fermi surfaces can change the Cooper pairing, leading to an SC-SC phase transition from the SC state under the nematic order to a higher-T c state (see Fig. 4b). The higher-T c state would provide an intriguing stage for the superconductivity mechanism common to iron-based superconductors with a high T c . However, such a theoretically predicted Lifshitz transition has not been reported so far because of the entangled P-T phase diagram and the experimental difficulties faced in observing Fermi surfaces under pressure.
In this study, we conducted 77 Se(I=1/2)-NMR measurements under pressure, focusing on 12% S-substituted FeSe, where the overlap of nematicity and magnetism is absent in the intermediate-pressure regime between 1 and 4 GPa 13,14 . Based on the results, we suggest that the theoretically predicted Lifshitz transition is observed as an anomaly of the DOS and as double SC domes accompanied by different types of AF fluctuations.
Typical NMR spectra corresponding to the nematic and magnetic orders are shown in Fig. 2a. We applied a magnetic field of 6.02 T parallel to the a axis in the tetragonal phase throughout the NMR measurements. In the nematic phase, the NMR spectra exhibit a double-edge structure 20 , as shown in the left panel of Fig. 2a. The double edges have been observed as two separate peaks for pure FeSe 18,19,22,23 . This edge structure disappears above 0.57 GPa. The spectra above 1 GPa exhibit a single peak. At 3.9 GPa, the 77 Se signal disappears at approximately 60 K because of the AF order (see the right panel of Fig. 2a). The T dependence of the linewidth at ambient pressure, 3.5 GPa and 3.9 GPa is shown in Fig. 2b. The AF order is observed via a remarkable increase in the linewidth and the loss of the signal. We defined T N as the temperature of the onset of linewidth broadening. Now, we focus on the Knight shift (K) in a paramagnetic state. Figure 3a shows the T dependence of K in a paramagnetic state. We adopted the average of the two edges for K in the nematic phase. The data below 3 GPa were already publisged in an early work 20 . The Knight shift above 3 GPa is T dependent even at low temperatures suggesting the influence of AF fluctuations, whereas the influence is absent below 3 GPa. Hereafter, we discuss the P dependence of K below 3 GPa in relation to the DOS shown in Fig. 1b. Fig 3b shows the P dependence of the NMR spectra at 60 K, and each spectrum is fitted by a Gaussian function. From the peak positions in Fig. 3b, the P dependence of K is obtained, as shown in Fig. 3c. Note that the anomaly at 1 GPa is observed at entire temperatures and therefore is not directly caused by the nematic transition.
The Knight shift in a paramagnetic state is decomposed as where K spin and K orb represent the spin and orbital parts of the Knight shift, respectively. The former and latter are T-dependent and T-independent, respectively. Experimentally, K is decomposed into K spin and K orb using the uniform spin susceptibility, χ(0) . The orbital part K orb is estimated to be 0.26% at ambient pressure [Supplemental material], which is almost the same as that obtained for pure FeSe 24 . The results suggest that K orb is insensitive to S substitution. The spin part K spin and χ(0) are related to K spin = Aχ(0) , where A is the hyperfine coupling.
(1) www.nature.com/scientificreports/ The monotonic decrease in K below 3 GPa with decreasing T suggests that the influence of magnetism is absent at low temperatures. In this case, χ(0) can be described using the formula for conventional paramagnetic metals and is related to the DOS of free electrons, D(E F ) . Therefore, K spin is proportional to D(E F ): Although the DOS shown in Fig. 1b is derived using a tight-binding (TB) model as described later, overall features can be roughly explained by the DOS for two-dimensional free electron systems. For two-dimensional free electron systems, D(E F ) is expressed as where N 2 , a, and m are the total number of lattices, lattice constant, and electron mass, respectively. The P dependence of D(E F ) originates only from that of a 2 . According to X-ray analyses up to 1 GPa 13 , the lattice constant (a) shrinks linearly with increasing pressure. The lattice constant also shrinks for S substitution: 30% S-substitution is equivalent to a pressure application of 1 GPa. Therefore, the discrepancy in a between pure FeSe and 12% S-substituted FeSe is trivial. We use a for pure FeSe because a for 12% S-substituted FeSe is not available at present. In addition, the data above 1 GPa is not available at present, and instead we adopted the extrapolation of the data below 1 GPa. The values of a 2 and K spin /a 2 normalized by those at ambient pressure are shown in Fig. 3d. The normalized K spin /a 2 is a quantity compared with the theoretical results shown in Fig. 1b. The step-like enhancement at 1 GPa reaches 10% of K spin /a 2 at ambient pressure, which is consistent with the theoretical calculation of the total DOS shown in Fig. 1b. The enhancement of K spin /a 2 seems to be smaller than that shown in Fig. 1b, implying that the size of the hole pocket at the point M is fairly small, as described later. We determined K spin at low pressures below 1 GPa, assuming that K orb is estimated to be ∼ 0.26% . However, at high pressures such as 2 or 3 GPa, the determination of K spin is very difficult because the data of χ(0) under pressure are not available. The assumption of K orb ∼ 0.26% leads to an unrealistic result, namely, K spin or the DOS at high pressures becomes lower than that at ambient pressure. To overcome this difficulty, we focus on a remarkable drop in K spin below T c at 2 and 3 GPa (see Fig. 3a). The apparent drop at 2 and 3 GPa originates from K spin , indicating that T-independent K orb decreases at high pressures. Therefore, we assumed that the decrease in K orb at high pressures is equivalent to the drop in K spin below T c . In Fig. 3d, we estimated the decrease in K orb to be 0.005 and 0.01 % for 2 and 3 GPa, respectively, from the drop in K spin below T c .
(2) K spin ∝ D(E F ).   www.nature.com/scientificreports/ The step-like enhancement is theoretically explained by the hole pocket at ( π , π ) appearing across the Fermi level owing to the lift of the d xy orbital (see Fig. 1a). Assuming that a=1, the P dependence of the DOS for each orbital was calculated for 10% S-substituted FeSe (Fig. 1b). The DOS was calculated using the TB models constructed from first-principles calculations based on the crystal structure. We denote the Hamiltonian for FeS x Se 1−x at pressure P (GPa) as H (0) x (P) . The model Hamiltonian used for these calculations is expressed as where E is the correction term added to fit the real size of the Fermi surfaces observed experimentally from ARPES and dHvA quantum oscillation [5][6][7][8][9][10] . Given that �H(P) ≡ H  The DOS enhancement appears at around 1.5 GPa, which is consistent with the experimental results of K shown in Fig. 2d. www.nature.com/scientificreports/ The DOS enhancement can affect low-energy magnetic fluctuations obtained from the relaxation rates divided by temperature, 1/T 1 T , as the nesting condition between Fermi surfaces changes. Figure 4a shows the T dependence of 1/T 1 T at several pressures. T c s shown by arrows were determined from the AC susceptibility measurements at 6.02 T [Supplemental material]. When the wave vector (q)-dependence of the hyperfine interaction is neglected, 1/T 1 T is expressed as follows: where ω and χ(q, ω) represent the NMR frequency and the dynamical spin susceptibility, respectively. Below T c , the signal intensity became extremely small, and thus, we could not detect signals below 10 K at 2.0 GPa and 15 K at 3.0 GPa, respectively. At ambient pressure, 1/T 1 T shows Curie-Weiss-like behavior in the nematic phase, indicating the development of AF fluctuations 25,26 : where a and b are independent of T. θ is estimated to be almost zero at ambient pressure. However, the Curie-Weiss behavior is strongly suppressed even at 0.57 GPa. Although the Curie-Weiss behavior is not clearly observed at pressures between 2 and 3.5 GPa, an anomaly of 1/T 1 T is observed at around T c . In this pressure regime, T c at 6.02 T gradually recovers with increasing pressure. The data at 3.9 GPa are completely different from those at 2.0 and 3.0 GPa in that the anomaly occurs not at T c but at T N . The data series in Fig. 4a is presented as a color plot in Fig. 4b. As shown in Fig. 4b, different types of AF fluctuations are observed in the P-T phase diagram, indicating that the origins of the AF fluctuations are different between the lower and higher pressure regimes. This result indicates a change in the nesting condition and confirms the theoretically predicted pressureinduced Lifshitz transition.
The results of K and 1/T 1 T show that the DOS and the AF fluctuations change at around 1 GPa, respectively. Interestingly, the AF fluctuations, which are unambiguous in the low-pressure regime where the nematic order occurs, unexpectedly become ambiguous in the high-pressure regime despite the AF phase boundary. In general, the Curie-Weiss behavior should be clearly observed near the AF phase boundary. Therefore, ambiguous AF fluctuations at high pressures are extremely rare compared to those of conventional AF magnets. This peculiarity www.nature.com/scientificreports/ arises because the nesting condition is not optimal, which implies that the d xy hole pocket is fairly small. Such a small hole pocket is consistent with the small increase in the DOS at 1 GPa. Another peculiar phenomenon is the loss of the NMR signal at low temperatures in the high-pressure regime above 1 GPa. This peculiarity can be attributed to the close relationship between the nematic and AF orders 21 . Nematic and/or AF states can appear at any pressure. Thus, a short-range AF order can develop even in the absence of a long-range AF order. A short-range AF order would make the NMR signal very weak and undetectable at low T below T c . A long-range AF order at pressures above 3.9 GPa can develop together with a finite order parameter from the short-range AF order.
We demonstrated the strong suppression of 1/T 1 T under pressure. A similar suppression is also observed by isovalent S substitution 27 , despite the fact that the Fermi surfaces become larger and the nesting condition becomes better with increasing S concentration [28][29][30] . S substitution would have the same effect as the application of pressure, because the atomic radius of S is smaller than that of Se. However, the chalcogen height decreases with increasing S concentration, in contrast to the application of pressure 13 . For the heavily S-substituted regime over 20%, where the BCS-BEC crossover has been suggested [31][32][33][34][35] , the Curie-Weiss behavior of 1/T 1 T is strongly suppressed, similar to 1/T 1 T for 12% S-substituted FeSe at 1 GPa. Although the strong suppression of the Curie-Weiss behavior is common, it is not clear whether the present high-pressure regime is smoothly linked with the heavily S-substituted regime. To solve this problem, further investigation is needed.
In conclusion, we performed 77 Se-NMR measurements on 12% S-substituted FeSe under pressures of up to 3.9 GPa. We observed the anomalies of K and 1/T 1 T corresponding to the theoretically predicted pressureinduced Lifshitz transition. These results indicate that nematicity and magnetism exhibit cooperative coupling. The AF fluctuation unambiguously develops in the nematic phase as the Curie-Weiss behavior of 1/T 1 T in the low-pressure regime below 1 GPa. In contrast, in the high-pressure regime between 1 and 3.9 GPa where the nematic order is absent, the AF fluctuation is strongly suppressed. Further, a high T c is realized in a pressure regime where the nematic order is absent and the correlated AF fluctuation is fairly weak. The emergence of the d xy orbital and its intra-orbital coupling play a key role for the high-T c superconductivity.

Methods
We performed 77 Se-NMR measurements at 6.02 T using a single crystal of 12% S-substituted FeSe with dimensions of approximately 1.0mm × 1.0mm × 0.5mm . We applied a magnetic field parallel to the FeSe planes to suppress the decrease in T c . We applied a pressure up to 3.9 GPa using a NiCrAl piston-cylinder-type pressure cell. The highest pressure attainable by clamping this pressure cell is 3.7 GPa because a decrease of 10% in pressure is inevitable after releasing a load. To attain a pressure of 3.9 GPa, we maintained a constant load by employing an oil press mounted on top of the cryostat 36 . We performed pulsed-NMR measurements using a conventional spectrometer and measured the relaxation time ( T 1 ) via the saturation-recovery method.

Data availability
Data are available from the corresponding author upon reasonable request.